9 MEASURABLE FUNCTIONS 153 Deduce that if a is coarser than jS, then H(a/y) g H(j8/y), and in particular H(a) ^ H(j8). (b) With the same notation, show that if j8 is coarser than y then H(a//3) g; H(a/y) (use the fact that the function t*-+t log t is convex in [0, 1]). In particular, we have H(a) J> H(a/y). For arbitrary a, j5, y, we have H((a V jS)/y) g H(a/y) + and in particular H(a V jS) g H(a) + 28. Let X be a compact space with measure /x(X) = 1, and let u : X~* X be a ^-measurable mapping such that w(/x) = /x (Problem 24). Let a be a finite partition of X into inte- grable subsets; if a = (A*)i «k^m , we denote by «~M(a) the partition formed by the sets (a) With the notation of Problem 27, show that the limit B-i V w~J( /H-l \ exists and is finite. (If 0n = H V «~"J(a) , show by the use of Problem 27(b) that V=° / <3m+rt ^ am + an , and then copy the proof of (1 5,2.7(i)). The number h(u, a) is called the entropy ofu relative to a. Show that h(u, a) fS H(a) (use Problem 27). (b) If a, ]8 are two finite partitions of X into integrable subsets, show that h(u, a V )8) g /z(w, a) + /z(w, j8) %, a) ^ A(w, ]3) + H(a/j8). If also a is coarser than j8, then A(w, a) g /z(w, j8). (Use Problem 27.) (c) Show that (Prove by induction on n that u,x)= limH(a/[ V «"</(a)M. n-.cc \ / \Jml JJ that H/V u~J(*)\ = H(a) + "E H(a/(fcVi «-*<«))V by using Problem 27.) (d) Show that, for each integer m jj> 1, (Observe that the right-hand side is equal to ( \ al 1 /m+n-l \ im-H( V «~J(a) -*oo ?Z \ J = 0 / and use Problem 27.) If in addition u is bijective and u~l is /^-measurable, then forthe " average" of the information corresponding to a.